Consider the rational number which in decimal form is .12345345345345...
It begins with two non-repeating digits followed by a block of 3 digits which repeats.

For the generalized repeating decimal .[a][b][b][b][b]...
Where [a] has n digits and [b] has m digits.

Find a general way to transform it into a rational number of the form p/r.

Take it in two parts:

.[a] is a/10^n

.[b][b][b][b]... would be b/(10^m - 1), but we have this divided by 10^n.

So the total value is a / 10^n + b / ((10^m - 1)*(10^n))

A common denominator is ((10^m - 1)*(10^n)), so we get

(a * (10^m - 1) + b) / ((10^m - 1)*(10^n))

In the specific example of .12345345345345..., a = 12, n = 2, b = 345, m = 3, and we get

(12 * (10^3 - 1) + 345) / ((10^3 - 1)*(10^2)) or

(12*999 + 345) / 99900 = 12333/99900

The numerator and denominator have a gcd of 3, so this reduces to 4111/33300.

Note that 10^m - 1 is a sequence of m 9's, and multiplying by 10^n is the same as tacking on n 0's to the end, for the mechanical purposes of forming the fraction.

Posted by Charlie on 2007-09-26 10:37:45